Space-time adaptive solution of fluid-structure interaction problems

Abstract

The treatment of fluid flow problems can be carried out effectively by least-squares first-order methods, which are an alternative to the usual formulations, because they always lead to symmetric positive definite matrices and do not need to satisfy the LBB condition. The first-order least-squares methods reformulate all the partial differential equations into a mathematically equivalent first-order system—by introducing additional variables. This formulation, when used together with a normal Galerkin formulation for a linear elastic solid, can be used to model the fluid-structure interaction problems. In both parts, a space-time finite element discretization is used. This allows the application of adaptive algorithms ,which simultaneously refine in space and time direction, and thus control the spatial and temporal discretisation error locally. Except one part of the coupling, the complete system of the new formulation for instationary fluid-structure interaction problems is found to be symmetric. This prevents one from using effective iterative solvers because the space-time adaptivity transfers the original pseudo 3D problem on a time slab consisting of one element in the time direction into a real 3D problem, which, in consequence, brings the direct solver to an end quickly.